We are meant to assume that 2006 is not a leap year. The problem came when I tried and didn't get an integer...
My answer so far is 2nd July....

I'm going to use a much smaller "year" as a demonstration here.

Think of it this way, say the year has 9 days in it. What is the middle day of the year?

Well, we have days
1 2 3 4 5 6 7 8 9
By inspection it looks like the middle day is 5. But . So how do we approach this?

Well, what you did amounts to solving , which gives a noninteger day. The reason for this is that 365 isn't even. But even then the middle day of an even list has no middle! For example
1 2 3 4 5 6 7 8
What is the middle number? There isn't one!

What this amounts to is that it is better to solve the equation . This will give you the correct day number. You can easily verify that this gives you the correct solution when the list has 9 elements.